Devices for multiplicative inverse calculation in the binary Galois fields

Abstract

Currently, the mathematical basis for digital signature processing is elliptic curves. In this case, the processing of the points of the elliptic curve is based on the operations in the Galois Field GF(2m), the field elements can be represented in polynomial and normal bases. There are two common methods for performing division in a Galois Field GF(2m). Method I is the extended Euclidean algorithm which uses a polynomial basis representation for GF(2m). This algorithm produces the quotient directly. Method II is exponentiation. The method efficiency improvements are especially significant when squaring can be done quickly (e.g., in a normal basis representation). The disadvantage of the extended Euclidean algorithm is the dependence of Galois Fields multiplicative inverse computing time on the value of operands. So, in the work some other undependable on operands methods are tested to select ones with the best hardware and time complexity for the polynomial basis. Three tested methods are based in exponentiation and one is direct division method. All methods were implemented and tested in FPGA.